Harder–Narasimhan recursion conjecture for higher asymptotic ranges

For each $d$, let $P_d(y)$ be the Poincaré polynomial of the corresponding moduli space, let $\widetilde P_{d,\chi}(y)$ be the Poincaré series of the stack of semistable sheaves, and let $\mathsf{HN}_k$ consist of pairs $(\boldsymbol d,\boldsymbol\chi)$ with $\boldsymbol d=(d_1,\ldots,d_m)$, $\boldsymbol\chi=(\chi_1,\ldots,\chi_m)$, $d_1+\cdots+d_m\leq k$, and $0\leq\chi_1/d_1<\cdots<\chi_m/d_m<3$. Set $d_0=d-\sum_i d_i$ and $s(\boldsymbol d)=\sum_{0\leq i<j\leq m}d_i d_j$. Higher-range Harder–Narasimhan conjecture. For $k\geq0$ and $d>k+1$,

$$\sum_{(\boldsymbol d,\boldsymbol\chi)\in\mathsf{HN}_k}y^{s(\boldsymbol d)}P_{d_0}\widetilde P_{d_1,\chi_1}\cdots\widetilde P_{d_m,\chi_m}=H(y)\pmod{y^{(k+1)(d-k-1)}}.$$

The conjecture is proposed as a generalization of the paper's higher asymptotic formulas and is not resolved in the source.